Calculation of Beam-Loaded Q in High-Power Klystrons

Instabilities in the gun region of a high-power klystron can occur when there is positive feedback between a mode and an induced current on the quasi-steady state beam emitted by the gun cathode[1]. This instability is dependent on the gun voltage, and is predicted on the basis of a negative total Q. The established method for computing the beam-loaded Q of a cavity involves using a time-dependent electromagnetic particle-in-cell (PIC) code to track beam particles through the quasi-static gun fields perturbed by the electromagnetic fields of a cavity eigenmode[2]. The energy imparted to the beam by the mode is obtained by integrating the Lorentz force along the particle tracks, and this quantity is simply related to the beam-loaded Q. We have developed an alternative approach that yields comparable accuracy but is computationally much simpler. The new method is based on a time-independent electrostatic PIC calculation, resulting in much faster solutions without loss of accuracy. We will present the theory and implementation of the new method, as well as benchmarks and results from analysis of the XP-4 klystron that show a potential instability near 3 GHz.